Social Statistics: Factorial ANOVA

When to analysis of variance with more than one factor

Main and interaction effects ToolPak

Review

2

Factorial Design: More than one factor (IV) is manipulated in the same experiment

This can produce main effects of either factor, and an interaction effect between the factors

This is the most comprehensive design, since factors interact with one another to produce behavior in the real world

The downside…you need far more subjects, time, and effort

Factorial Designs

3

Main effect: Mean differences along the levels of one factor (one-way F-ratio)

In addition to the two factors alone, we can evaluate mean differences that result from unique combinations of the two factors.

An interaction between two factors occurs whenever mean differences between individual treatment conditions (combinations of two factors) are different from the overall mean effects of the factors

“The effects of one factor vary as a function of the other”

Main Effects and Interactions

4

Two-factor ANOVA will do three things: Examine differences in sample means

for humidity (factor A) Examine differences in sample means

for temperature (factor B) Examine differences in sample means

for combinations of humidity and temperature (factor A and B).

Three sets of hypotheses and three F-ratios.

Hypotheses and F-ratios

5

Two factors: gender (male or female) and treatment (high or low impact) The same people experience both the

high and low impact conditions

Example

Weight loss Treatment

High Impact Low Impact

Gender Female

Male

6

Three questions: Is there a difference between the levels

of impact (main effect)? Is there a difference between the two

levels of gender (main effect)? What is the effect of difference levels of

impact for males or females (interaction effects)

Example

7

high-impact male

high-impact female

low-impact male

low-impact female

76 65 88 6578 90 76 6776 65 76 6776 90 76 8776 65 56 7874 90 76 5674 90 76 5476 79 98 5676 70 88 5455 90 78 56

Data

Two-way ANOVA or factorial ANOVA

8

Null hypothesis

Research hypothesis

Hypotheses

lowhighH :0 femalemaleH :0

femalelowmalelowfemalehighmalehighH :0

lowhighH :1 femalemaleH :1

femalelowmalelowfemalehighmalehighH :1

9

Excel Toolpak

10

Anova: Two-Factor With Replication

SUMMARY high-impact low-impact Totalmale      

Count 10 10 20Sum 737 788 1525Average 73.7 78.8 76.25Variance 44.45555556 121.9555556 85.67105

female      

Count 10 10 20Sum 794 640 1434Average 79.4 64 71.7Variance 141.3777778 126.2222222 189.1684

Total      

Count 20 20Sum 1531 1428Average 76.55 71.4Variance 96.57631579 175.2

ANOVASource of Variation SS df MS F P-value F crit

Sample 207.025 1 207.025 1.908016 0.175698 4.113165Columns 265.225 1 265.225 2.444407 0.126693 4.113165Interaction 1050.625 1 1050.625 9.682932 0.003631 4.113165Within 3906.1 36 108.5028

Total 5428.975 39

Excel Toolpak

1. There is no main effect for treatment or gender (p=0.127, 0.176)

2. There is interaction effect (p=0.004)

3. It does not matter if you are in the high or low impact treatment group, or if you are male or female

4. It does matter if you are in both conditions simultaneously the treatment does have an impact differentially on the weight loss of males than on females

11